Let $R$ be the region in the first quadrant enclosed by the graphs of $y = 2x$ and $y = x^{2}$, as shown in the figure above. (a) Find the area of $R$. (b) The region $R$ is the base of a solid. For this solid, at each $x$ the cross section perpendicular to the $x$-axis has area $A(x) = \sin\left(\frac{\pi}{2} x\right)$. Find the volume of the solid. (c) Another solid has the same base $R$. For this solid, the cross sections perpendicular to the $y$-axis are squares. Write, but do not evaluate, an integral expression for the volume of the solid.
Let $R$ be the region in the first quadrant enclosed by the graphs of $y = 2x$ and $y = x^{2}$, as shown in the figure above.
(a) Find the area of $R$.
(b) The region $R$ is the base of a solid. For this solid, at each $x$ the cross section perpendicular to the $x$-axis has area $A(x) = \sin\left(\frac{\pi}{2} x\right)$. Find the volume of the solid.
(c) Another solid has the same base $R$. For this solid, the cross sections perpendicular to the $y$-axis are squares. Write, but do not evaluate, an integral expression for the volume of the solid.